Research Paper
Volume : 4 | Issue : 12 | December 2015 • ISSN No 2277 - 8179
Solution of Geometric Construction
Problems with using the Programs the
Geometer's Sketchpad
Education
KEYWORDS : Information culture;
Information and communication technology (ICT); Computer animation.
Kaskatayeva Bahytkul
Rahimzhanovna
Doctor of pedagogical Sciences, associate Professor, associate Professor of Kazakh State
Womens Teacher Training University
A.B.Kokazhaeva
associate Professor of Kazakh State Womens Teacher Training University
Z.M.Toleuhanova
Senior Lecturer of Kazakh State Womens Teacher Training University
A.U.Dauletkulova
associate Professor of Kazakh State Womens Teacher Training University
ABSTRACT
The article describes the content and methodology of solving geometric problems on the build with the help of
the program «The Geometer's Sketchpad». The possibilities of the program in the field of analysis, research, build,
evidence. Outlines the benefits of the program compared with traditional style perform geometric constructions.
Introduction.
The Geometer’s Sketchpad® is the world’s leading software for
teaching mathematics. Sketchpad® gives students at all levels—
from third grade through college—a tangible, visual way to learn
mathematics that increases their engagement, understanding,
and achievement. Make math more meaningful and memorable
using Sketchpad.
Elementary students can manipulate dynamic models of fractions, number lines, and geometric patterns. Middle school students can build their readiness for algebra by exploring ratio and
proportion, rate of change, and functional relationships through
numeric, tabular, and graphical representations. And high school
students can use Sketchpad to construct and transform geometric shapes and functions—from linear to trigonometric—promoting deep understanding.
Sketchpad is the optimal tool for interactive whiteboards.
Teachers can use it daily to illustrate and illuminate mathematical ideas. Classroom-tested activities are accompanied by presentation sketches and detailed teacher notes, which provide
suggestions for use by teachers as a demonstration tool or for
use by students in a computer lab or on laptops.
For students the information and communication technologies
are both the object of study and as a tool of subject and pedagogical activity, and as a means of training and methodological
support of educational process in the school.
This program is invaluable with the first steps in the study of geometry.
A special place in geometry take the task of drawing, which is
difficult in its decision, the build. The lesson typically, such tasks
spent a lot of time, the quality of the builds students is very low.
It is quite another matter, if such tasks to solve in the«The Geometer’s Sketchpad». All measurements are accurate, build fast,
and students and teachers care only about the algorithm for
solving the problem, which is then checked with the source data
changes.
The effectiveness of mathematical problems and exercises largely depends on the degree of creative activity of students in their
decision.
In fact, one of the main task assignments and exercises is to
strengthen the mental activity of students in the classroom.
Geometric problems on the build first of all, awakens the
thought of students, makes it work, develops, improves. Speaking about enhancing the thinking of the disciples, we must not
forget that when solving geometric problems on drawing students not only perform the build, conversion and remember the
wording, but are also given clear thinking, ability to reason, to
compare and contrast facts, find similarities and differences, to
draw correct conclusions.
The problem of preparing students for the development of information culture of students stems with increasing the functional
value of information ... [1].
The four basic steps of solving geometric problems (analysis,
drawing, proof, research) we have added a fifth: after the problem has been solved, you should create a computer animation
illustrating and explaining the process of solving problems.
Preparation of future teachers of mathematics has been and remains difficult. Today the computer just works wonders, using
its vast possibilities. Today has created many different training
programs. These include
When drawing a computer animation one of the possible ways
is using «The Geometer’s Sketchpad».Tasks on the build - tasks,
in which with the help of a compass, ruler and elementary geometric constructions are built geometric shapes.
computer program «The Geometer’s Sketchpad», authored by
Nicholas Jackiw.
Methodology of solving geometric problems on the build
with the help.
During this work we will try to perform basic geometric constructions are not using the traditional tools for drawing and
using computer technology, namely the Russian version of the
popular American training program on the geometry of «The
Geometer’s Sketchpad» (Russian version of “Living geometry”),
developed by Key Curriculum Press.
«The Geometer’s Sketchpad» is a set of tools that provides all
the necessary tools for constructing drawings and their research. It allows you to open and verify geometric facts. The program allows you to “liven up”
the drawings, smoothly changing the position of the initial point.
390
IJSR - INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH
Research Paper
Objective: the solution of some problems on the build using
«The Geometer’s Sketchpad».
Objectives:
1. Solve some problems on the build using “Live geometry”.
2. To explore possible solutions to these problems.
3. To analyze the features of the program “Living geometry when
performing geometric constructions.
Geometric construction is a way of solving the problem in which
the answer is get a graphical way. Build perform the drawing
tools with maximum precision and accuracy, since it depends
on the accuracy of the solution. The geometry section, which examines geometric constructions, called constructive geometry.
The main concept of constructive geometry is the concept to
construct the geometric figure. This concept is accepted without
definition, the specific meaning known from practice, where it
means: to draw, to hold (line), mark (point) [2].
Construction problem consists in the fact that these instruments are required to construct the figure, if given some other
figure and shows some relations between the elements of the
desired shape. Each figure satisfying task, called the solution of
tasks.
Find the solution of geometric construction problems - specify
an ending sequence of basic structures, then the desired shape
is considered to be built by the axiom of constructive geometry.
A list of the main builds, and, consequently, the progress of solutions depend on a common set of tools. It should be noted that
this approach is in determining location decisions are not rational. Sometimes it is better to enlarge the build steps.
Considered as a build step, whole blocks of the main builds.
These blocks represent the solution of elementary problems on
the build. Going to call them elementary constructions. Then we
can give the following definition.
Solve the problem in the drawing - it means to indicate the end
sequence of basic and elementary drawing, after which the desired pattern can be considered built in force general axioms of
constructive geometry. Sometimes the conditions of the problem to build to can be a few figures. To solve the problem - it
means to find all its solutions. Let us explain this definition.
Shapes satisfying the condition of the tasks vary in size, shape
and position on the plane. Shapes, differing in size or shape, will
be considered different. With a location this is the case.
Problem is considered solved if:
a) the figures F1 , F2 ,... constructed satisfy the problem;
b) it is proved that any form that satisfies the problem
statement coincides with one of them.
This objective may have different solutions [3].
The stages of the build
When solving problems main difficulty is the question of how to
find the solution. The resolution of this question will be easier if
you follow a certain scheme of reasoning. This scheme consists
of four phases: analysis, construction, proof, research. Note that
this classical scheme is not, of course, necessary and immutable.
1. Analysis. In the analysis a search for solving the problem as
follows: assume the problem is solved, built (by hand) the
desired shape attach to it the data based on those relations
that are specified in the problem statement. Notice that the
construction of the desired shape f is reduced to the construction of the other figures F1 , drawing F1 reduces to
Volume : 4 | Issue : 12 | December 2015 • ISSN No 2277 - 8179
the construction of F2, and so on, After a finite number of
steps, you can come to some figure of Fn, the construction
of which is known.
If the auxiliary drawing will not be able to find the progress
of solutions, it is advisable to enter into a drawing additional shapes to make additional construction, to make geometric transformations, etc.
2. The drawing consists of specifying a finite sequence of basic
builds (or previously solved problems), which is sufficient to
produce the desired shape was built.
The drawing is usually accompanied by graphics of each
step with the help of these tools.
3. The evidence is intended to establish that built figure indeed satises tasks.
The proof is based on the assumption that each construction step can be performed.
4. Research. When the analysis is usually limited to finding
any solution, offering the possibility of building activities.
For a complete solution to find out:
1) you can always use your favorite way;
2) how to build the desired shape when the method is not suitable;
3) How many solutions has the task
These questions form the content of the study. Thus, the study aims to - to create conditions for the solubility and determine the number of solutions.
We discuss each step of the construction and the possibility of
uniqueness.
The question remains: if there are other solutions . Solution to
this problem coincides with one of the already obtained solutions [4].
We will show the solution of tasks on the drawing with the use
of package “Living geometry”.
Problem solving ancient geometric problems, which are addressed in the school geometry course. Difficulties in solving
such problems a lot:
1) finding ways to solve the problem by linking the desired
item and data of the problem;
2) the execution of construction;
3) proof of the correctness of execution of structures;
4) study of the problem, that is, asking the question of whether the data for the solution of any problem, and if so, how
many solutions.
Thus, in mathematical class not so much to solve such problems. Yes, and the process of building with a pencil, ruler, compass very inaccurate.
Another thing, the solution of such problems in the computer
program “Live geometry”, in which to build a model for such
problems, you can always.
Decision of geometrical problems with a program «The Geometer’s Sketchpad».
Task 1.
Construction of the triangle on the two sides and the angle between them. Analysis: The problem reduces to the construction
of an angle equal to the given angle. Construction of a ruler and
compass:
1. Construct an angle equal to the given angle.
2. On the sides of the angle to postpone the length specified
interval.
IJSR - INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH
391
Research Paper
Volume : 4 | Issue : 12 | December 2015 • ISSN No 2277 - 8179
3. The points are connected by a straight line.
Given the two sides of a predetermined length and angle between them.
The conclusion: the problem has a unique solution if the angle
is acute, obtuse, or straight. If the angle is deployed, the problem
has no solution.
Task2.
Build a triangle. Given three sides.
Analysis using the ruler, you can build a ray AB and a segment of
a given length. (Figure 3)
Figure 2
B
[AL) and its segment AB = c of a given length.
Construct a circle with center at A And radius AC = b.
Construct a circle with center at B And radius BC = a.
Drawing on «The Geometer’s Sketchpad» ( Figure 4):
The problem has two solutions;
Study: 1) a + b > с (Figure 5)
Research: 2) a + b ≤ с (Figure 5)
A
Conclusion:
The triangle can be built, if the sum of two sides is greater third
party. Thus is formed the ability of future mathematics teachers
use in their work, computer information technology.
Figure 3
C
Geometer’s Sketchpad, allows you to easily create scalable and
editable drawings, to make all necessary measurements. The
program provides activities in the field of analysis, research, construction, evidence of problem solving.
Sketchpad is convenient to measure length, area, and angles
with the selected precision, to create tens of teaching and research «live» drawings. Using the program you can also find examples, manual
search which would take a lot of time or simply impossible. Opportunities to work with the program «The Geometer’s
Sketchpad» is extremely varied. Judicious use of the program
has advantages over the traditional style of performing
geometric constructions.
Figure 4
Figure 5
Figure 1
REFERENCE
[1] Kaskatayeva B. R. Integration of innovative methods of training. International scientific conference. Russian Journal: "Geometry and Geometrical Education at the Modern High and Higher School", Tolyatti. Nov. 2014, | [2] Semenov E. E. Study geometry: KN. for students 6-8 class. environments. SHK. “Education”, Moskva, 46-56, 1987. | [3] Elective course in mathematics: textbook for 7-9 grade environments. SHK. / Comp. I. L. Nicholas. - M.: Education, 1991. - S. 66-85.
| [4] Shuvalov E. H., Kaplun Century. And. Geometry: Textbook. the manual for the preparatory departments of universities. - M.: Higher. school, 1980. - S. 31-33. | [5] The Geometer's
Sketchpad. RuTracker.org rutracker.org/forum/viewtopic.... | [6] Geometry, 7-9: textbook. for omeopatia. institutions / HP, Athanasian, V. F. Butuzov and others): Education, 2004. 168.
||
392
IJSR - INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH